Skew-orthogonal Polynomials Research Articles

A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type

A Riemann-Hilbert Approach to Skew-Orthogonal Polynomials of Symplectic Type

Skew-Orthogonal Polynomials and Pfaff Lattice Hierarchy Associated With an Elliptic Curve

Abstract Starting with a skew-symmetric inner product over an elliptic curve, we propose the concept of elliptic skew-orthogonal polynomials. Inspired by the Landau–Lifshitz hierarchy and its corresponding time evolutions, we obtain the recurrence relation and the $\tau $-function representation for such a novel class of skew-orthogonal polynomials. Furthermore, a bilinear integral identity is derived through the so-called Cauchy–Stieljes transformation, from which we successfully establish the connection between the elliptic skew-orthogonal polynomials and an elliptic extension of the Pfaff lattice hierarchy.

Asymptotic expansion of matrix models in the multi-cut regime

Abstract We establish the asymptotic expansion in $\beta $ matrix models with a confining, off-critical potential in the regime where the support of the equilibrium measure is a finite union of segments. We first address the case where the filling fractions of these segments are fixed and show the existence of a $\frac {1}{N}$ expansion. We then study the asymptotics of the sum over the filling fractions to obtain the full asymptotic expansion for the initial problem in the multi-cut regime. In particular, we identify the fluctuations of the linear statistics and show that they are approximated in law by the sum of a Gaussian random variable and an independent Gaussian discrete random variable with oscillating center. Fluctuations of filling fractions are also described by an oscillating discrete Gaussian random variable. We apply our results to study the all-order small dispersion asymptotics of solutions of the Toda chain associated with the one Hermitian matrix model ( $\beta = 2$ ) as well as orthogonal ( $\beta = 1$ ) and skew-orthogonal ( $\beta = 4$ ) polynomials outside the bulk.

Christoffel transformations for (partial-)skew-orthogonal polynomials and applications

In this article, we consider the Christoffel transformations for skew-orthogonal polynomials and partial-skew-orthogonal polynomials. We demonstrate that the Christoffel transformations can act as spectral problems for discrete integrable hierarchies, and therefore we derive certain integrable hierarchies from these transformations. Some reductional cases are also considered.

Multiple Skew-Orthogonal Polynomials and 2-Component Pfaff Lattice Hierarchy

Multiple Skew-Orthogonal Polynomials and 2-Component Pfaff Lattice Hierarchy

Universal scaling limits of the symplectic elliptic Ginibre ensemble

We consider the eigenvalues of symplectic elliptic Ginibre matrices which are known to form a Pfaffian point process whose correlation kernel can be expressed in terms of the skew-orthogonal Hermite polynomials. We derive the scaling limits and the convergence rates of the correlation functions at the real bulk/edge of the spectrum, which in particular establishes the local universality at strong non-Hermiticity. Furthermore, we obtain the subleading corrections of the edge correlation kernels, which depend on the non-Hermiticity parameter contrary to the universal leading term. Our proofs are based on the asymptotic behavior of the complex elliptic Ginibre ensemble due to Lee and Riser as well as on a version of the Christoffel–Darboux identity, a differential equation satisfied by the skew-orthogonal polynomial kernel.

Skew-Orthogonal Polynomials in the Complex Plane and Their Bergman-Like Kernels

Non-Hermitian random matrices with symplectic symmetry provide examples for Pfaffian point processes in the complex plane. These point processes are characterised by a matrix valued kernel of skew-orthogonal polynomials. We develop their theory in providing an explicit construction of skew-orthogonal polynomials in terms of orthogonal polynomials that satisfy a three-term recurrence relation, for general weight functions in the complex plane. New examples for symplectic ensembles are provided, based on recent developments in orthogonal polynomials on planar domains or curves in the complex plane. Furthermore, Bergman-like kernels of skew-orthogonal Hermite and Laguerre polynomials are derived, from which the conjectured universality of the elliptic symplectic Ginibre ensemble and its chiral partner follow in the limit of strong non-Hermiticity at the origin. A Christoffel perturbation of skew-orthogonal polynomials as it appears in applications to quantum field theory is provided.

Tracy-Widom Distributions for the Gaussian Orthogonal and Symplectic Ensembles Revisited: A Skew-Orthogonal Polynomials Approach

We study the distribution of the largest eigenvalue in the "Pfaffian" classical ensembles of random matrix theory, namely in the Gaussian orthogonal (GOE) and Gaussian symplectic (GSE) ensembles, using semi-classical skew-orthogonal polynomials, in analogue to the approach of Nadal and Majumdar (NM) for the Gaussian unitary ensemble (GUE). Generalizing the techniques of Adler, Forrester, Nagao and van Moerbeke, and using "overlapping Pfaffian" identities due to Knuth, we explicitly construct these semi-classical skew-orthogonal polynomials in terms of the semi-classical orthogonal polynomials studied by NM in the case of the GUE. With these polynomials we obtain expressions for the cumulative distribution functions of the largest eigenvalue in the GOE and the GSE. Further, by performing asymptotic analysis of these skew-orthogonal polynomials in the limit of large matrix size, we obtain an alternative derivation of the Tracy-Widom distributions for GOE and GSE. This asymptotic analysis relies on a certain Pfaffian identity, the proof of which employs the characterization of Pfaffians in terms of perfect matchings and link diagrams.

Preserving topology while breaking chirality: from chiral orthogonal to anti-symmetric Hermitian ensemble

We consider a parameter dependent ensemble of two real random matrices with Gaussian distribution. It describes the transition between the symmetry class of the chiral Gaussian orthogonal ensemble (Cartan class B|DI) and the ensemble of antisymmetric Hermitian random matrices (Cartan class B|D). It enjoys the special feature that, depending on the matrix dimension , it has exactly zero-mode for even (odd), throughout the symmetry transition. This ‘topological protection’ is reminiscent of properties of topological insulators. We show that our ensemble represents a Pfaffian point process which is typical for such transition ensembles. On a technical level, our results follow from the applicability of the Harish-Chandra integral over the orthogonal group. The matrix-valued kernel determining all eigenvalue correlation functions is explicitly constructed in terms of skew-orthogonal polynomials, depending on the topological index . These polynomials interpolate between Laguerre and even (odd) Hermite polynomials for , in terms of which the two limiting symmetry classes can be solved. Numerical simulations illustrate our analytical results for the spectral density and an expansion for the distribution of the smallest eigenvalue at finite .

Partial-Skew-Orthogonal Polynomials and Related Integrable Lattices with Pfaffian Tau-Functions

Skew-orthogonal polynomials (SOPs) arise in the study of the n-point distribution function for orthogonal and symplectic random matrix ensembles. Motivated by the average of characteristic polynomials of the Bures random matrix ensemble studied in [22], we propose the concept of partial-skew-orthogonal polynomials (PSOPs) as a modification of the SOPs, and then the PSOPs with a variety of special skew-symmetric kernels and weight functions are addressed. By considering appropriate deformations of the weight functions, we derive nine integrable lattices in different dimensions. As a consequence, the tau-functions for these systems are shown to be expressed in terms of Pfaffians and the wave vectors PSOPs. In fact, the tau-functions also admit the representations of multiple integrals. Among these integrable lattices, some of them are known, while the others are novel to the best of our knowledge. In particular, one integrable lattice is related to the partition function of the Bures random matrix ensemble. Besides, we derive a discrete integrable lattice, which can be used to compute certain vector Pad\'e approximants. This yields the first example regarding the connection between integrable lattices and vector Pad\'e approximants, for which Hietarinta, Joshi and Nijhoff pointed out that "This field remains largely to be explored." in the recent monograph [27, Section 4.4] .

GUE-chGUE transition preserving chirality at finite matrix size

We study a random matrix model which interpolates between the singular values of the Gaussian unitary ensemble (GUE) and of the chiral Gaussian unitary ensemble (chGUE). This symmetry crossover is analogous to the one realized by the Hermitian Wilson Dirac operator in lattice QCD, but our model preserves chiral symmetry of chGUE exactly unlike the Hermitian Wilson Dirac operator. This difference has a crucial impact on the statistics of near-zero eigenvalues, though both singular value statistics build a Pfaffian point process. The model in the present work is motivated by the Dirac operator of 3D staggered fermions, 3D QCD at finite isospin chemical potential, and 4D QCD at high temperature. We calculate the spectral statistics at finite matrix dimension. For this purpose, we derive the joint probability density of the singular values, the skew-orthogonal polynomials and the kernels for the k-point correlation functions. The skew-orthogonal polynomials are constructed by the method of mixing bi-orthogonal and skew-orthogonal polynomials, which is an alternative approach to Mehta’s one. We compare our results with Monte Carlo simulations and study the limits to chGUE and GUE. As a side product, we also calculate a new type of a unitary group integral.

An induced real quaternion spherical ensemble of random matrices

We study the induced spherical ensemble of non-Hermitian matrices with real quaternion entries (considering each quaternion as a [Formula: see text] complex matrix). We define the ensemble by the matrix probability distribution function that is proportional to [Formula: see text] These matrices can also be constructed via a procedure called ‘inducing’, using a product of a Wishart matrix (with parameters [Formula: see text]) and a rectangular Ginibre matrix of size [Formula: see text]. The inducing procedure imposes a repulsion of eigenvalues from [Formula: see text] and [Formula: see text] in the complex plane with the effect that in the limit of large matrix dimension, they lie in an annulus whose inner and outer radii depend on the relative size of [Formula: see text], [Formula: see text] and [Formula: see text]. By using functional differentiation of a generalized partition function, we make use of skew-orthogonal polynomials to find expressions for the eigenvalue [Formula: see text]-point correlation functions, and in particular the eigenvalue density (given by [Formula: see text]). We find the scaled limits of the density in the bulk (away from the real line) as well as near the inner and outer annular radii, in the four regimes corresponding to large or small values of [Formula: see text] and [Formula: see text]. After a stereographic projection, the density is uniform on a spherical annulus, except for a depletion of eigenvalues on a great circle corresponding to the real axis (as expected for a real quaternion ensemble). We also form a conjecture for the behavior of the density near the real line based on analogous results in the [Formula: see text] and [Formula: see text] ensembles; we support our conjecture with data from Monte Carlo simulations of a large number of matrices drawn from the [Formula: see text] induced spherical ensemble. This ensemble is a quaternionic analog of a model of a one-component charged plasma on a sphere, with soft wall boundary conditions.

Real eigenvalue statistics for products of asymmetric real Gaussian matrices

Random matrices formed from i.i.d. standard real Gaussian entries have the feature that the expected number of real eigenvalues is non-zero. This property persists for products of such matrices, independently chosen, and moreover it is known that as the number of matrices in the product tends to infinity, the probability that all eigenvalues are real tends to unity. We quantify the distribution of the number of real eigenvalues for products of finite size real Gaussian matrices by giving an explicit Pfaffian formula for the probability that there are exactly k real eigenvalues as a determinant with entries involving particular Meijer G-functions. We also compute the explicit form of the Pfaffian correlation kernel for the correlation between real eigenvalues, and the correlation between complex eigenvalues. The simplest example of these — the eigenvalue density of the real eigenvalues — gives by integration the expected number of real eigenvalues. Our ability to perform these calculations relies on the construction of certain skew-orthogonal polynomials in the complex plane, the computation of which is carried out using their relationship to particular random matrix averages.

The smallest eigenvalue distribution in the real Wishart–Laguerre ensemble with even topology

We consider rectangular random matrices of size belonging to the real Wishart–Laguerre ensemble also known as the chiral Gaussian orthogonal ensemble. This ensemble appears in many applications like QCD, mesoscopic physics, and time series analysis. We are particularly interested in the distribution of the smallest non-zero eigenvalue and the gap probability to find no eigenvalue in an interval . While for odd topology explicit closed results are known for finite and infinite matrix size, for even only recursive expressions in p are available. The smallest eigenvalue distribution as well as the gap probability for general even ν is equivalent to expectation values of characteristic polynomials raised to a half-integer power. The computation of such averages is done via a combination of skew-orthogonal polynomials and bosonisation methods. The results are given in terms of Pfaffian determinants both at finite p and in the hard edge scaling limit ( and ν fixed) for an arbitrary even topology ν. Numerical simulations for the correlated Wishart ensemble illustrate the universality of our results in this particular limit. These simulations point to a validity of the hard edge scaling limit beyond the invariant case.

The singular values of the GOE

As a unifying framework for examining several properties that nominally involve eigenvalues, we present a particular structure of the singular values of the Gaussian orthogonal ensemble (GOE): the even-location singular values are distributed as the positive eigenvalues of a Gaussian ensemble with chiral unitary symmetry, while the odd-location singular values, conditioned on the even-location ones, can be algebraically transformed into a set of independent χ-distributed random variables. We discuss three applications of this structure: first, there is a pair of bidiagonal square matrices, whose singular values are jointly distributed as the even- and odd-location ones of the GOE; second, the magnitude of the determinant of the GOE is distributed as a product of simple independent random variables; third, on symmetric intervals, the gap probabilities of the GOE can be expressed in terms of the Laguerre unitary ensemble. We work specifically with matrices of finite order, but by passing to a large matrix limit, we also obtain new insight into asymptotic properties such as the central limit theorem of the determinant or the gap probabilities in the bulk-scaling limit. The analysis in this paper avoids much of the technical machinery (e.g. Pfaffians, skew-orthogonal polynomials, martingales, Meijer G-function, etc.) that was previously used to analyze some of the applications.

The Interpolating Airy Kernels for the $$\beta =1$$ β = 1 and $$\beta =4$$ β = 4 Elliptic Ginibre Ensembles

We consider two families of non-Hermitian Gaussian random matrices, namely the elliptical Ginibre ensembles of asymmetric N-by-N matrices with Dyson index beta=1 (real elements) and with beta=4 (quaternion-real elements). Both ensembles have already been solved for finite N using the method of skew-orthogonal polynomials, given for these particular ensembles in terms of Hermite polynomials in the complex plane. In this paper we investigate the microscopic weakly non-Hermitian large-N limit of each ensemble in the vicinity of the largest or smallest real eigenvalue. Specifically, we derive the limiting matrix-kernels for each case, from which all the eigenvalue correlation functions can be determined. We call these new kernels the "interpolating" Airy kernels, since we can recover -- as opposing limiting cases -- not only the well-known Airy kernels for the Hermitian ensembles, but also the complementary error function and Poisson kernels for the maximally non-Hermitian ensembles at the edge of the spectrum. Together with the known interpolating Airy kernel for beta=2, which we rederive here as well, this completes the analysis of all three elliptical Ginibre ensembles in the microscopic scaling limit at the spectral edge.

Products of independent quaternion Ginibre matrices and their correlation functions

We discuss the product of independent induced quaternion (β = 4) Ginibre matrices, and the eigenvalue correlations of this product matrix. The joint probability density function for the eigenvalues of the product matrix is shown to be identical to that of a single Ginibre matrix, but with a more complicated weight function. We find the skew-orthogonal polynomials corresponding to the weight function of the product matrix, and use the method of skew-orthogonal polynomials to compute the eigenvalue correlation functions for product matrices of finite size. The radial behavior of the density of states is studied in the limit of large matrices, and the macroscopic density is discussed. The microscopic limit at the origin, at the edge(s) and in the bulk is also discussed for the radial behavior of the density of states.

Skew orthogonal polynomials for the real and quaternion real Ginibre ensembles and generalizations

There are some distinguished ensembles of non-Hermitian random matrices for which the joint probability density function can be written down explicitly, are unchanged by rotations, and furthermore which have the property that the eigenvalues form a Pfaffian point process. For these ensembles, in which the elements of the matrices are either real, or real quaternion, the kernel of the Pfaffian is completely determined by certain skew orthogonal polynomials, which permit an expression in terms of averages over the characteristic polynomial, and the characteristic polynomial multiplied by the trace. We use Schur polynomial theory, knowledge of the value of a Schur polynomial averaged against real, and real quaternion Gaussian matrices, and the Selberg integral to evaluate these averages.

Mixing of orthogonal and skew-orthogonal polynomials and its relation to Wilson RMT

The unitary Wilson random matrix theory is an interpolation between the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble. This new way of interpolation is also reflected in the orthogonal polynomials corresponding to such a random matrix ensemble. Although the chiral Gaussian unitary ensemble and the Gaussian unitary ensemble are associated with the Dyson index β = 2, the intermediate ensembles exhibit a mixing of orthogonal polynomials and skew-orthogonal polynomials. We consider the Hermitian and the non-Hermitian Wilson random matrix and derive the corresponding polynomials, their recursion relations, Christoffel–Darboux-like formulas, Rodrigues formulas and representations as random matrix averages in a unifying way. With the help of these results, we derive the unquenched k-point correlation function of the Hermitian and the non-Hermitian Wilson random matrix in terms of two-flavor partition functions only. This representation is due to a Pfaffian factorization. It drastically simplifies the expressions, which can be easily numerically evaluated. It also serves as a good starting point for studying the Wilson–Dirac operator in the ϵ-regime of lattice quantum chromodynamics.

Discrete Spectral Transformations of Skew Orthogonal Polynomials and Associated Discrete Integrable Systems

Discrete spectral transformations of skew orthogonal polynomials are presented. From these spectral transformations, it is shown that the corresponding discrete integrable systems are derived both in 1+1 dimension and in 2+1 dimension. Especially in the (2+1)-dimensional case, the corresponding system can be extended to 2x2 matrix form. The factorization theorem of the Christoffel kernel for skew orthogonal polynomials in random matrix theory is presented as a by-product of these transformations.